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Determination of the SPHARA basis functions for an EEG sensor setup
Introduction
A Fourier basis can be obtained as a solution to Laplace’s eigenvalue problem
where \(\mathbf{L}\) is a discrete Laplace–Beltrami operator in matrix notation, the eigenvectors \(\boldsymbol{\varphi}_i\) contain the spatial harmonic functions and the eigenvalues \(\lambda_i \ge 0\) are real-valued. When \(\mathbf{L}\) is obtained from a finite element (FEM) discretization of the Laplace–Beltrami operator, the quantities \(\sqrt{\lambda_i}\) can be interpreted as spatial angular frequencies; see SPHARA – The theoretical background in a nutshell for details.
By solving a Laplace eigenvalue problem, it is also possible to determine a basis for a spatial Fourier analysis. Often in practical applications, a measured quantity to be subjected to a Fourier analysis is only known at spatially discrete sampling points (the sensor positions). An arbitrary arrangement of sample points on a surface in three-dimensional space can be described by means of a triangular mesh. In the case of an EEG system, the sample positions (the vertices of the triangular mesh) are the locations of the sensors arranged on the head surface. A SPHARA basis is a solution of a Laplace eigenvalue problem for the given triangle mesh, that can be obtained by discretizing a Laplace–Beltrami operator for the mesh and solving the Laplace eigenvalue problem in equation (1). The SpharaPy package provides three methods for the discretization of the Laplace–Beltrami operator; unit weighting of the edges and weighting with the inverse of the Euclidean distance of the edges, and a FEM approach. For more detailed information please refer to SPHARA – The theoretical background in a nutshell and Eigensystems of discrete Laplace–Beltrami operators.
At the beginning we import three modules of the SpharaPy package as well as several other packages and single functions of packages.
# Code source: Uwe Graichen
# License: BSD 3 clause
# import modules from spharapy package
# import additional modules used in this tutorial
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.mplot3d import Axes3D # noqa: F401 (registers 3D)
import spharapy.datasets as sd
import spharapy.spharabasis as sb
import spharapy.trimesh as tm
Specification of the spatial configuration of the EEG sensors
Import information about EEG sensor setup of the sample data set
In this tutorial we will determine a SPHARA basis for a 256 channel
EEG system with equidistant layout. The data set is one of
the example data sets contained in the SpharaPy toolbox, see
spharapy.datasets and
spharapy.datasets.load_eeg_256_channel_study().
# loading the 256 channel EEG dataset from spharapy sample datasets
mesh_in = sd.load_eeg_256_channel_study()
The dataset includes lists of vertices, triangles, and sensor labels, as well as EEG data from previously performed experiment addressing the cortical activation related to somatosensory-evoked potentials (SEP).
print(mesh_in.keys())
dict_keys(['vertlist', 'trilist', 'labellist', 'eegdata'])
The triangulation of the EEG sensor setup consists of 256 vertices and 480 triangles.
vertlist = np.array(mesh_in["vertlist"])
trilist = np.array(mesh_in["trilist"])
print("vertices = ", vertlist.shape)
print("triangles = ", trilist.shape)
vertices = (256, 3)
triangles = (482, 3)
fig = plt.figure()
fig.subplots_adjust(left=0.02, right=0.98, top=0.98, bottom=0.02)
ax = fig.add_subplot(111, projection="3d")
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.set_title("The triangulated EEG sensor setup")
ax.view_init(elev=20.0, azim=80.0)
ax.set_aspect("auto")
ax.plot_trisurf(
vertlist[:, 0],
vertlist[:, 1],
vertlist[:, 2],
triangles=trilist,
color="lightblue",
edgecolor="black",
linewidth=0.5,
shade=True,
)
plt.show()
Create a SpharaPy TriMesh instance
In the next step we create an instance of the class
spharapy.trimesh.TriMesh from the list of vertices and
triangles.
The class spharapy.trimesh.TriMesh provides a
number of methods to determine certain properties of the triangle
mesh required to generate the SPHARA basis, listed below:
# print all implemented methods of the TriMesh class
print([func for func in dir(tm.TriMesh) if not func.startswith("__")])
['adjacent_tri', 'is_edge', 'laplacianmatrix', 'massmatrix', 'one_ring_neighborhood', 'remove_vertices', 'stiffnessmatrix', 'trilist', 'vertlist', 'weightmatrix']
Determining SPHARA bases using different discretisation approaches
Computing the basis functions
In the final step of the tutorial we will calculate SPHARA bases for
the given EEG sensor setup. For this we create three
instances of the class spharapy.spharabasis.SpharaBasis. We
use the three discretization approaches implemented in this class
for the Laplace–Beltrami operator: unit weighting (‘unit’) and
inverse Euclidean weigthing (‘inv_euclidean’) of the edges of the
triangular mesh as well as the FEM discretization (‘fem’)
# 'unit' discretization
sphara_basis_unit = sb.SpharaBasis(mesh_eeg, "unit")
basis_functions_unit, natural_frequencies_unit = sphara_basis_unit.basis()
# 'inv_euclidean' discretization
sphara_basis_ie = sb.SpharaBasis(mesh_eeg, "inv_euclidean")
basis_functions_ie, natural_frequencies_ie = sphara_basis_ie.basis()
# 'fem' discretization
sphara_basis_fem = sb.SpharaBasis(mesh_eeg, "fem")
basis_functions_fem, natural_frequencies_fem = sphara_basis_fem.basis()
Visualization the basis functions
The first 15 spatially low-frequency SPHARA basis functions are shown below, starting with DC at the top left.
SPHARA basis using the discretization approache ‘unit’
figsb1, axes1 = plt.subplots(nrows=5, ncols=3, figsize=(8, 12), subplot_kw={"projection": "3d"})
for i in range(np.size(axes1)):
colors = np.mean(basis_functions_unit[trilist, i + 0], axis=1)
ax = axes1.flat[i]
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.view_init(elev=60.0, azim=80.0)
ax.set_aspect("auto")
trisurfplot = ax.plot_trisurf(
vertlist[:, 0],
vertlist[:, 1],
vertlist[:, 2],
triangles=trilist,
cmap=plt.cm.bwr,
edgecolor="white",
linewidth=0.0,
)
trisurfplot.set_array(colors)
trisurfplot.set_clim(-0.15, 0.15)
cbar = figsb1.colorbar(
trisurfplot,
ax=axes1.ravel().tolist(),
shrink=0.85,
orientation="horizontal",
fraction=0.05,
pad=0.05,
anchor=(0.5, -4.5),
)
plt.subplots_adjust(left=0.0, right=1.0, bottom=0.08, top=1.0)
plt.show()
SPHARA basis using the discretization approache ‘inv_euclidean’
figsb1, axes1 = plt.subplots(nrows=5, ncols=3, figsize=(8, 12), subplot_kw={"projection": "3d"})
for i in range(np.size(axes1)):
colors = np.mean(basis_functions_ie[trilist, i + 0], axis=1)
ax = axes1.flat[i]
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.view_init(elev=60.0, azim=80.0)
ax.set_aspect("auto")
trisurfplot = ax.plot_trisurf(
vertlist[:, 0],
vertlist[:, 1],
vertlist[:, 2],
triangles=trilist,
cmap=plt.cm.bwr,
edgecolor="white",
linewidth=0.0,
)
trisurfplot.set_array(colors)
trisurfplot.set_clim(-0.18, 0.18)
cbar = figsb1.colorbar(
trisurfplot,
ax=axes1.ravel().tolist(),
shrink=0.85,
orientation="horizontal",
fraction=0.05,
pad=0.05,
anchor=(0.5, -4.5),
)
plt.subplots_adjust(left=0.0, right=1.0, bottom=0.08, top=1.0)
plt.show()
SPHARA basis using the discretization approache ‘fem’
figsb1, axes1 = plt.subplots(nrows=5, ncols=3, figsize=(8, 12), subplot_kw={"projection": "3d"})
for i in range(np.size(axes1)):
colors = np.mean(basis_functions_fem[trilist, i + 0], axis=1)
ax = axes1.flat[i]
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_zlabel("z")
ax.view_init(elev=60.0, azim=80.0)
ax.set_aspect("auto")
trisurfplot = ax.plot_trisurf(
vertlist[:, 0],
vertlist[:, 1],
vertlist[:, 2],
triangles=trilist,
cmap=plt.cm.bwr,
edgecolor="white",
linewidth=0.0,
)
trisurfplot.set_array(colors)
trisurfplot.set_clim(-0.01, 0.01)
cbar = figsb1.colorbar(
trisurfplot,
ax=axes1.ravel().tolist(),
shrink=0.85,
orientation="horizontal",
fraction=0.05,
pad=0.05,
anchor=(0.5, -4.5),
)
plt.subplots_adjust(left=0.0, right=1.0, bottom=0.08, top=1.0)
plt.show()
Total running time of the script: (0 minutes 2.811 seconds)